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Classification of quadrics

  1. May 21, 2008 #1
    Hi,

    can someone help me on a website for the classification of quadrics(ellipsoids,paraboloids,....) starting from the general second order equation.

    Thanks
     
  2. jcsd
  3. May 21, 2008 #2

    HallsofIvy

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  4. May 21, 2008 #3

    mathwonk

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  5. May 22, 2008 #4
    Hi,

    What I really mean is:

    you have a general equation of the second order in three variables and by translations

    and rotations to remove Xy-Xz and yz terms you transform it to the canonical forms

    of quadrics.(ellipsoids,hyperboloids,cone......)

    Bye
     
  6. May 22, 2008 #5

    mathwonk

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    do you want to know how to complete the square in a quadratic equation? thats all it is.
     
  7. May 22, 2008 #6

    mathwonk

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    in one variable you start from X^2 + bX + c, and you get

    X^2 + bX + (b/2)^2 +c - (b/2)^2 = (X+ b/2)^2 + [(4c-b^2)]/4.

    in 2 variables you start from X^2 + bXY + cY^2, and get

    X^2 + bXY +(bY/2)^2 + cY^2 - (bY/2)^2

    = (X+bY/2)^2 + [4c-b^2]/4 Y^2.

    so now you replace X+ bY/2 by W and you have W^2 + [4c-b^2]/4 Y^2.

    there are no WY terms, and so you can tell which quadric you have by the sign of

    [4c-b^2]/4.

    maybe you want to expand this a little, by throwing in a third letter Z, but this is the main trick.

    i guess for a general homogeneous function of three variables you want to diagonalize the matrix representing the quadric.

    the previous discussion above works for Z = quadratic in (X,Y), which arises in calculus of two variables.
     
    Last edited: May 22, 2008
  8. May 23, 2008 #7
    Hi,
    mathwonk

    yes that's.for three variables I think you have to diagonalize the matrix.First you you must

    compute the eigen-values of the matrix .

    Thank you for your help.
     
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